Operator enumeration
Which interactions do these fields allow? Enumerate every monomial through a chosen dimension, then count its independent Lorentz and gauge invariants.
Scope: 4D scalar and Weyl fields, no derivatives or field strengths, canonical dimension ≤ 6. Read the counting contract.
Enumerate the complete scalar/Weyl polynomial sectors through dimension six, with explicit conjugate fields. Derivatives and field strengths are excluded.
The first number sums invariant-space dimensions; the second counts field monomials with at least one invariant. Conjugate monomials are separate. Neither number is the count of real Lagrangian parameters after field redefinitions.
Allowed field monomials
| Δ | Field monomial | Complex invariants | Calculation |
|---|---|---|---|
| 2 | H · H† | 1 | Inspect H, H† |
| 4 | Q · uᶜ · H | 1 | Inspect Q, uᶜ, H |
| 4 | Q · dᶜ · H† | 1 | Inspect Q, dᶜ, H† |
| 4 | Q† · uᶜ† · H† | 1 | Inspect Q†, uᶜ†, H† |
| 4 | Q† · dᶜ† · H | 1 | Inspect Q†, dᶜ†, H |
| 4 | L · eᶜ · H† | 1 | Inspect L, eᶜ, H† |
| 4 | L† · eᶜ† · H | 1 | Inspect L†, eᶜ†, H |
| 4 | H × 2 · H† × 2 | 1 | Inspect H, H† |
| 5 | L × 2 · H × 2 | 1 | Inspect L, H |
| 5 | L† × 2 · H† × 2 | 1 | Inspect L†, H† |
| 6 | Q × 3 · L | 1 | Inspect Q, L |
| 6 | Q × 2 · Q† × 2 | 2 | Inspect Q, Q† |
| 6 | Q × 2 · uᶜ · dᶜ | 2 | Inspect Q, uᶜ, dᶜ |
| 6 | Q × 2 · uᶜ† · eᶜ† | 1 | Inspect Q, uᶜ†, eᶜ† |
| 6 | Q · Q† · uᶜ · uᶜ† | 2 | Inspect Q, Q†, uᶜ, uᶜ† |
| 6 | Q · Q† · dᶜ · dᶜ† | 2 | Inspect Q, Q†, dᶜ, dᶜ† |
| 6 | Q · Q† · L · L† | 2 | Inspect Q, Q†, L, L† |
| 6 | Q · Q† · eᶜ · eᶜ† | 1 | Inspect Q, Q†, eᶜ, eᶜ† |
| 6 | Q · uᶜ · L · eᶜ | 2 | Inspect Q, uᶜ, L, eᶜ |
| 6 | Q · uᶜ · H × 2 · H† | 1 | Inspect Q, uᶜ, H, H† |
| 6 | Q · uᶜ† · dᶜ† · L | 1 | Inspect Q, uᶜ†, dᶜ†, L |
| 6 | Q · dᶜ · L† · eᶜ† | 1 | Inspect Q, dᶜ, L†, eᶜ† |
| 6 | Q · dᶜ · H · H† × 2 | 1 | Inspect Q, dᶜ, H, H† |
| 6 | Q† × 3 · L† | 1 | Inspect Q†, L† |
| 6 | Q† × 2 · uᶜ · eᶜ | 1 | Inspect Q†, uᶜ, eᶜ |
| 6 | Q† × 2 · uᶜ† · dᶜ† | 2 | Inspect Q†, uᶜ†, dᶜ† |
| 6 | Q† · uᶜ · dᶜ · L† | 1 | Inspect Q†, uᶜ, dᶜ, L† |
| 6 | Q† · uᶜ† · L† · eᶜ† | 2 | Inspect Q†, uᶜ†, L†, eᶜ† |
| 6 | Q† · uᶜ† · H · H† × 2 | 1 | Inspect Q†, uᶜ†, H, H† |
| 6 | Q† · dᶜ† · L · eᶜ | 1 | Inspect Q†, dᶜ†, L, eᶜ |
| 6 | Q† · dᶜ† · H × 2 · H† | 1 | Inspect Q†, dᶜ†, H, H† |
| 6 | uᶜ × 2 · uᶜ† × 2 | 1 | Inspect uᶜ, uᶜ† |
| 6 | uᶜ × 2 · dᶜ · eᶜ | 1 | Inspect uᶜ, dᶜ, eᶜ |
| 6 | uᶜ · uᶜ† · dᶜ · dᶜ† | 2 | Inspect uᶜ, uᶜ†, dᶜ, dᶜ† |
| 6 | uᶜ · uᶜ† · L · L† | 1 | Inspect uᶜ, uᶜ†, L, L† |
| 6 | uᶜ · uᶜ† · eᶜ · eᶜ† | 1 | Inspect uᶜ, uᶜ†, eᶜ, eᶜ† |
| 6 | uᶜ† × 2 · dᶜ† · eᶜ† | 1 | Inspect uᶜ†, dᶜ†, eᶜ† |
| 6 | dᶜ × 2 · dᶜ† × 2 | 1 | Inspect dᶜ, dᶜ† |
| 6 | dᶜ · dᶜ† · L · L† | 1 | Inspect dᶜ, dᶜ†, L, L† |
| 6 | dᶜ · dᶜ† · eᶜ · eᶜ† | 1 | Inspect dᶜ, dᶜ†, eᶜ, eᶜ† |
| 6 | L × 2 · L† × 2 | 1 | Inspect L, L† |
| 6 | L · L† · eᶜ · eᶜ† | 1 | Inspect L, L†, eᶜ, eᶜ† |
| 6 | L · eᶜ · H · H† × 2 | 1 | Inspect L, eᶜ, H, H† |
| 6 | L† · eᶜ† · H × 2 · H† | 1 | Inspect L†, eᶜ†, H, H† |
| 6 | eᶜ × 2 · eᶜ† × 2 | 1 | Inspect eᶜ, eᶜ† |
| 6 | H × 3 · H† × 3 | 1 | Inspect H, H† |
No nonzero sectors occur at this dimension.
Inspect a sector to see its statistics, selection rules, and field-component dimensions in the local-operator workbench. The transfer carries the complete inventory and global quotient; additional charge filters are recorded in this enumeration.
Enumeration coverage and exact checks
All occupations within the published input bounds are considered. Exact charge, center, Lorentz-parity, and fermion-capacity tests may eliminate impossible candidates before character projection. Passing these tests alone never counts a monomial as an invariant.
Global descent for the whole inventory
| Species | Generator phase (mod 1) | Descent |
|---|---|---|
| Q | Descends | |
| Q† | Descends | |
| uᶜ | Descends | |
| uᶜ† | Descends | |
| dᶜ | Descends | |
| dᶜ† | Descends | |
| L | Descends | |
| L† | Descends | |
| eᶜ | Descends | |
| eᶜ† | Descends | |
| H | Descends | |
| H† | Descends |
Saved inventory and calculation files carry the full fields and selected charge rules. Loading a file validates its inputs and recomputes the result; cached answers are ignored. A physical setup v1 can also be imported: its chosen monomial is replaced by enumeration through the selected dimension.
Enumerate the fields. Project the full representation.
Each occupation vector defines a multigraded polynomial space. Scalars use symmetric powers and Weyl fields use exterior powers of the complete Lorentz × gauge × flavor representation. Exact singlet projection supplies its multiplicity. The workbench sums those dimensions over the selected monomials.
Defined here
Four-dimensional Lorentzian scalar and Weyl fields; SU(3) × SU(2) × U(1) with the specified diagonal quotients; dimension at most six; explicit conjugate variables; up to three additional integer or cyclic charge rules.
Further structure
Derivatives, field strengths, equations of motion, integration by parts, Hermiticity, flavor-basis equivalences, anomaly tests, and normalized contraction bases need additional calculations.
There are at most sixteen species and three flavors per species. A scalar has canonical dimension one and a Weyl field dimension three halves. The finite degree bounds and exact resource guards are stated in the method reference. A guard failure or cancellation returns no completed enumeration.
Enumeration method and bounds · From fields to interactions · Exact invariant-counting method
Primary sources and interpretation
- Efficient Hilbert Series for Effective Theories§II A, Eqs. (2)–(4): symmetric/exterior powers and Haar projection without derivatives.
- Hilbert Series for Constructing Lagrangians: expanding the phenomenologist’s toolbox§4.1, Eq. (4.2): 57 QQQL operators for three generations; Appendix A: characters, plethystic exponentials and Haar measure.
- Enumerating the operators of an effective field theory§3, Eq. (49) and Table 2: QQQL flavor count n_L n_Q (2 n_Q² + 1)/3; Eq. (51): n²(n²−1)/12 four-sterile-Weyl invariants.
- Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry§2 and §3.2: two-component Lorentz spinors, spinor contractions and symmetric fermion mass matrices.
- Higher dimension operators in the SM EFT§2: operator Hilbert series and the additional treatment required for derivatives, equations of motion and integration by parts.
- Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and PhenomenologyAppendix A and Table 2: flavor-dependent coefficient counts, including 30 and 2205 baryon-preserving four-fermion directions for one and three generations.
- Renormalization group evolution of dimension-six baryon number violating operatorsEq. (1) and Table I: four baryon-violating field structures and 273 complex coefficients in the three-generation ΔB = 1 sector without sterile neutrinos.
- Baryon Number, Lepton Number, and Operator Dimension in the Standard ModelEqs. (15)–(18): d ≡ (ΔB − ΔL)/2 modulo 2 for Standard Model Lorentz- and hypercharge-invariant operators.
Enumeration algorithm: operator-enumeration-1.0.0. Application 6.0.0.
Complex invariant counts include algebraic contraction identities. They do not by themselves specify a Hermitian action or establish anomaly cancellation. Computational checks and source evidence remain separate from independent human review.